Proving the Span of u_1, u_2, ..., u_k and Av

[Phymath]

Proof Help! Spans!

I was give the fallowing to prove and i would like it know if anyone thinks i did this right lol...

Let $$u_1, u_2, ..., u_k$$ be vectors in $$R^n$$ and $$A$$ be a $$m x n$$ matrix. Prove that if $$v$$ is int he span of $$u_1, u_2, ..., u_k$$ then $$Av$$ is in the span of $$Au_1, Au_2, ..., Au_k$$.

this is what i did to prove this...

If $$v$$ is in the span of the set $$S$$ which follows $$S = {u_1, u_2, ...,u_k}$$ then there is set of coeffients that make the following linear combonation of the elements of the set $$S$$ that follows the linear equation to prove that $$v$$ is in the span of $$S$$...
$$c_1 u_1 + c_2 u_2 + ... + c_k u_k = v$$

to show that $$Av$$ is in the span of $$Au_1, Au_2, ..., Au_k$$
thus the fallowing linear equation must be true...$$c_1 Au_1 + c_2 Au_2 + ... + c_k Au_k = Av$$ factoring matrix $$A$$ gives $$A(c_1 u_1 + c_2 u_2 + ... + c_k u_k) \rightarrow Av$$ thus proving Av is in the span of $$Au_1, Au_2, ..., Au_k$$

so did i make any mistakes is this crap? let me know please...

FOR ANYONE WHO'S LATEX ISN'T LOADING HERES THE RAW TYPING
I was give the fallowing to prove and i would like it know if anyone thinks i did this right lol...

Let u_1, u_2, ..., u_k be vectors in R^n and A be a m x n matrix. Prove that if v is int he span of u_1, u_2, ..., u_k then Av is in the span of Au_1, Au_2, ..., Au_k .

this is what i did to prove this...

If v is in the span of the set S which follows S = {u_1, u_2, ...,u_k} then there is set of coeffients that make the following linear combonation of the elements of the set S that follows the linear equation to prove that v is in the span of S...
c_1 u_1 + c_2 u_2 + ... + c_k u_k = v

to show that Av is in the span of Au_1, Au_2, ..., Au_k
thus the fallowing linear equation must be true...c_1 Au_1 + c_2 Au_2 + ... + c_k Au_k = Av factoring matrix A gives A(c_1 u_1 + c_2 u_2 + ... + c_k u_k) -> Av thus proving Av is in the span of Au_1, Au_2, ..., Au_k

so did i make any mistakes is this crap? let me know please...

 

[mathwonk]

just tell me briefly what you are trying to do. i cannot digest all that crap.

 

[matt grime]

the short answer is, yes, you've got it. (A is linear so it commutes with scalar multiplication and addition of vectors, that's all they're getting at in the question).

 

[mathwonk]

if v = some linear combination of the u's, then since matrix multiplication commutes with linear combinations, then Av is the same linear combination of the Au's. QED.